A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is

From a pack of 52 cards, we draw one by one, without replacement. If $f(n)$ is the probability that an Ace will appear at the $n^{\text{th}}$ turn, then

A and B take part in a rifle shooting match. The probability of A hitting the target is 0.4, while the probability of B hitting the target is 0.6. If A has the first shot, post which both strike alternately, then the probability that A hits the target before B hits it is

A man is known to speak the truth on an average 4 out of 5 times. He throws a die and reports that it is a five. The probability that it is actually a five is

The probability that a randomly chosen positive divisor of $10 ^ {2023}$ is an integer multiple of $10 ^ {2001}$ is

The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 0.8 is

A fair die is rolled repeatedly. The probability that the cumulative sum is at least 17 in the third trial is